EEE 498/598 Overview of Electrical Engineering

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EEE 498/598Overview of Electrical Engineering

• Lecture 12
• Overview Of Circuit Theory
• Lumped Circuit Elements Topology Of Circuits
  Resistors KCL and KVL Resistors in Series and
  Parallel Energy Storage Elements First-Order
  Circuits

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Lecture 12 Objectives

• To commence our study of circuit theory.
• To develop an understanding of the concepts of
  Lumped circuit elements topology of circuits
  resistors KCL and KVL resistors in series and
  parallel energy storage elements and
  first-order circuits.

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Overview of Circuit Theory

• Electrical circuit elements are idealized models
  of physical devices that are defined by
  relationships between their terminal voltages and
  currents. Circuit elements can have two or more
  terminals.
• An electrical circuit is a connection of circuit
  elements into one or more closed loops.

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Overview of Circuit Theory

• A lumped circuit is one where all the terminal
  voltages and currents are functions of time only.
  Lumped circuit elements include resistors,
  capacitors, inductors, independent and dependent
  sources.
• An distributed circuit is one where the terminal
  voltages and currents are functions of position
  as well as time. Transmission lines are
  distributed circuit elements.

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Overview of Circuit Theory

• Basic quantities are voltage, current, and power.
• The sign convention is important in computing
  power supplied by or absorbed by a circuit
  element.
• Circuit elements can be active or passive active
  elements are sources.

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Overview of Circuit Theory

• Current is moving positive electrical charge.
• Measured in Amperes (A) 1 Coulomb/s
• Current is represented by I or i.
• In general, current can be an arbitrary function
  of time.
• Constant current is called direct current (DC).
• Current that can be represented as a sinusoidal
  function of time (or in some contexts a sum of
  sinusoids) is called alternating current (AC).

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Overview of Circuit Theory

• Voltage is electromotive force provided by a
  source or a potential difference between two
  points in a circuit.
• Measured in Volts (V) 1 J of energy is needed to
  move 1 C of charge through a 1 V potential
  difference.
• Voltage is represented by V or v.

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Overview of Circuit Theory

• The lower case symbols v and i are usually used
  to denote voltages and currents that are
  functions of time.
• The upper case symbols V and I are usually used
  to denote voltages and currents that are DC or AC
  steady-state voltages and currents.

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Overview of Circuit Theory

• Current has an assumed direction of flow
  currents in the direction of assumed current flow
  have positive values currents in the opposite
  direction have negative values.
• Voltage has an assumed polarity volt drops in
  with the assumed polarity have positive values
  volt drops of the opposite polarity have negative
  values.
• In circuit analysis the assumed polarity of
  voltages are often defined by the direction of
  assumed current flow.

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Overview of Circuit Theory

• Power is the rate at which energy is being
  absorbed or supplied.
• Power is computed as the product of voltage and
  current
• Sign convention positive power means that energy
  is being absorbed negative power means that
  power is being supplied.

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Overview of Circuit Theory

• If p(t) gt 0, then the circuit element is
  absorbing power from the rest of the circuit.
• If p(t) lt 0, then the circuit element is
  supplying power to the rest of the circuit.

i(t)
Rest of circuit

v(t)
-
Circuit element under consideration
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Overview of Circuit Theory

• If power is positive into a circuit element, it
  means that the circuit element is absorbing
  power.
• If power is negative into a circuit element, it
  means that the circuit element is supplying
  power. Only active elements (sources) can supply
  power to the rest of a circuit.

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Active and Passive Elements

• Active elements can generate energy.
• Examples of active elements are independent and
  dependent sources.
• Passive elements cannot generate energy.
• Examples of passive elements are resistors,
  capacitors, and inductors.
• In a particular circuit, there can be active
  elements that absorb power for example, a
  battery being charged.

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Independent and Dependent Sources

• An independent source (voltage or current) may
  be DC (constant) or time-varying its value does
  not depend on other voltages or currents in the
  circuit.
• A dependent source has a value that depends on
  another voltage or current in the circuit.

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Independent Sources
Voltage Source
Current Source
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Dependent Sources
vf(vx)
vf(ix)


-
-
Voltage Controlled Voltage Source (VCVS)
Current Controlled Voltage Source (CCVS)
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Dependent Sources
If(Vx)
If(Ix)
Voltage Controlled Current Source (VCCS)
Current Controlled Current Source (CCCS)
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Passive Lumped Circuit Elements

• Resistors
• Capacitors
• Inductors

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Topology of Circuits

• A lumped circuit is composed of lumped elements
  (sources, resistors, capacitors, inductors) and
  conductors (wires).
• All the elements are assumed to be lumped, i.e.,
  the entire circuit is of negligible dimensions.
• All conductors are perfect.

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Topology of Circuits

• A schematic diagram is an electrical
  representation of a circuit.
• The location of a circuit element in a schematic
  may have no relationship to its physical
  location.
• We can rearrange the schematic and have the same
  circuit as long as the connections between
  elements remain the same.

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Topology of Circuits

• Example Schematic of a circuit

Ground a reference point where the voltage (or
potential) is assumed to be zero.
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Topology of Circuits

• Only circuit elements that are in closed loops
  (i.e., where a current path exists) contribute to
  the functionality of a circuit.

This circuit element can be removed without
affecting functionality. This circuit behaves
identically to the previous one.
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Topology of Circuits

• A node is an equipotential point in a circuit.
  It is a topological concept in other words,
  even if the circuit elements change values, the
  node remains an equipotential point.
• To find a node, start at a point in the circuit.
  From this point, everywhere you can travel by
  moving only along perfect conductors is part of a
  single node.

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Topology of Circuits

• A loop is any closed path through a circuit in
  which no node is encountered more than once.
• To find a loop, start at a node in the circuit.
  From this node, travel along a path back to the
  same node ensuring that you do not encounter any
  node more than once.
• A mesh is a loop that has no other loops inside
  of it.

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Topology of Circuits

• If we know the voltage at every node of a
  circuit relative to a reference node (ground),
  then we know everything about the circuit i.e.,
  we can determine any other voltage or current in
  the circuit.
• The same is true if we know every mesh current.

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Topology of Circuits

• In this example there are 5 nodes and 2 meshes.
• In addition to the meshes, there is one
  additional loop (following the outer perimeter of
  the circuit).

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Resistors

• A resistor is a circuit element that dissipates
  electrical energy (usually as heat).
• Real-world devices that are modeled by resistors
  incandescent light bulb, heating elements
  (stoves, heaters, etc.), long wires
• Parasitic resistances many resistors on circuit
  diagrams model unwanted resistances in
  transistors, motors, etc.

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Resistors

• Resistance is measured in Ohms (W)
• The relationship between terminal voltage and
  current is governed by Ohms law
• Ohms law tells us that the volt drop in the
  direction of assumed current flow is Ri

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KCL and KVL

• Kirchhoffs Current Law (KCL) and Kirchhoffs
  Voltage Law (KVL) are the fundamental laws of
  circuit analysis.
• KCL is the basis of nodal analysis in which
  the unknowns are the voltages at each of the
  nodes of the circuit.
• KVL is the basis of mesh analysis in which the
  unknowns are the currents flowing in each of the
  meshes of the circuit.

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KCL and KVL

• KCL
• The sum of all currents entering a node is zero,
  or
• The sum of currents entering node is equal to sum
  of currents leaving node.

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KCL and KVL

• KVL
• The sum of voltages around any loop in a circuit
  is zero.

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KCL and KVL

• In KVL
• A voltage encountered to - is positive.
• A voltage encountered - to is negative.
• Arrows are sometimes used to represent voltage
  differences they point from low to high voltage.

v(t)
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Resistors in Series

• A single loop circuit is one which has only a
  single loop.
• The same current flows through each element of
  the circuit - the elements are in series.

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Resistors in Series

• Two elements are in series if the current that
  flows through one must also flow through the
  other.

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Resistors in Series

• Consider two resistors in series with a voltage
  v(t) across them

Voltage division
v1(t)
v2(t)
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Resistors in Series

• If we wish to replace the two series resistors
  with a single equivalent resistor whose
  voltage-current relationship is the same, the
  equivalent resistor has a value given by

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Resistors in Series

• For N resistors in series, the equivalent
  resistor has a value given by

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Resistors in Parallel

• When the terminals of two or more circuit
  elements are connected to the same two nodes, the
  circuit elements are said to be in parallel.

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Resistors in Parallel

• Consider two resistors in parallel with a voltage
  v(t) across them

Current division
i(t)

i1(t)
i2(t)
R1
R2
v(t)
-
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Resistors in Parallel

• If we wish to replace the two parallel resistors
  with a single equivalent resistor whose
  voltage-current relationship is the same, the
  equivalent resistor has a value given by

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Resistors in Parallel

• For N resistors in parallel, the equivalent
  resistor has a value given by

R3
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Energy Storage Elements

• Capacitors store energy in an electric field.
• Inductors store energy in a magnetic field.
• Capacitors and inductors are passive elements
• Can store energy supplied by circuit
• Can return stored energy to circuit
• Cannot supply more energy to circuit than is
  stored.

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Energy Storage Elements

• Voltages and currents in a circuit without energy
  storage elements are solutions to algebraic
  equations.
• Voltages and currents in a circuit with energy
  storage elements are solutions to linear,
  constant coefficient differential equations.

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Energy Storage Elements

• Electrical engineers (and their software tools)
  usually do not solve the differential equations
  directly.
• Instead, they use
• LaPlace transforms
• AC steady-state analysis
• These techniques covert the solution of
  differential equations into algebraic problems.

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Energy Storage Elements

• Energy storage elements model electrical loads
• Capacitors model computers and other electronics
  (power supplies).
• Inductors model motors.
• Capacitors and inductors are used to build
  filters and amplifiers with desired frequency
  responses.
• Capacitors are used in A/D converters to hold a
  sampled signal until it can be converted into
  bits.

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Capacitors

• Capacitance occurs when two conductors are
  separated by a dielectric (insulator).
• Charge on the two conductors creates an electric
  field that stores energy.
• The voltage difference between the two conductors
  is proportional to the charge.
• The proportionality constant C is called
  capacitance.
• Capacitance is measured in Farads (F).

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Capacitors
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Capacitors

• The voltage across a capacitor cannot change
  instantaneously.
• The energy stored in the capacitors is given by

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Inductors

• Inductance occurs when current flows through a
  (real) conductor.
• The current flowing through the conductor sets up
  a magnetic field that is proportional to the
  current.
• The voltage difference across the conductor is
  proportional to the rate of change of the
  magnetic flux.
• The proportionality constant is called the
  inductance, denoted L.
• Inductance is measured in Henrys (H).

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Inductors
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Inductors

• The current through an inductor cannot change
  instantaneously.
• The energy stored in the inductor is given by

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Analysis of Circuits Containing Energy Storage
Elements

• Need to determine
• The order of the circuit.
• Forced (particular) and natural
  (complementary/homogeneous) responses.
• Transient and steady state responses.
• 1st order circuits - the time constant.
• 2nd order circuits - the natural frequency and
  the damping ratio.

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Analysis of Circuits Containing Energy Storage
Elements

• The number and configuration of the energy
  storage elements determines the order of the
  circuit.
• n ? of energy storage elements
• Every voltage and current is the solution to a
  differential equation.
• In a circuit of order n, these differential
  equations are linear constant coefficient and
  have order n.

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Analysis of Circuits Containing Energy Storage
Elements

• Any voltage or current in an nth order circuit is
  the solution to a differential equation of the
  form
• as well as initial conditions derived from the
  capacitor voltages and inductor currents at t
  0-.

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Analysis of Circuits Containing Energy Storage
Elements

• The solution to any differential equation
  consists of two parts
• v(t) vp(t) vc(t)
• Particular (forced) solution is vp(t)
• Response particular to the source
• Complementary/homogeneous (natural) solution is
  vc(t)
• Response common to all sources

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Analysis of Circuits Containing Energy Storage
Elements

• The particular solution vp(t) is typically a
  weighted sum of f(t) and its first n derivatives.
• If f(t) is constant, then vp(t) is constant.
• If f(t) is sinusoidal, then vp(t) is sinusoidal.

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Analysis of Circuits Containing Energy Storage
Elements

• The complementary solution is the solution to
• The complementary solution has the form

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Analysis of Circuits Containing Energy Storage
Elements

• s1 through sn are the roots of the characteristic
  equation

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Analysis of Circuits Containing Energy Storage
Elements

• If si is a real root, it corresponds to a
  decaying exponential term
• If si is a complex root, there is another complex
  root that is its complex conjugate, and together
  they correspond to an exponentially decaying
  sinusoidal term

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Analysis of Circuits Containing Energy Storage
Elements

• The steady state (SS) response of a circuit is
  the waveform after a long time has passed.
• DC SS if response approaches a constant.
• AC SS if response approaches a sinusoid.
• The transient response is the circuit response
  minus the steady state response.

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Analysis of Circuits Containing Energy Storage
Elements

• Transients usually are associated with the
  complementary solution.
• The actual form of transients usually depends on
  initial capacitor voltages and inductor currents.
• Steady state responses usually are associated
  with the particular solution.

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First-Order Circuits

• Any circuit with a single energy storage element,
  an arbitrary number of sources, and an arbitrary
  number of resistors is a circuit of 1st order.
• Any voltage or current in such a circuit is the
  solution to a 1st order differential equation.

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First-Order Circuits

• Examples of 1st order circuits
• Computer RAM
• A dynamic RAM stores ones as charge on a
  capacitor.
• The charge leaks out through transistors modeled
  by large resistances.
• The charge must be periodically refreshed.

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First-Order Circuits

• Examples of 1st order circuits (Contd)
• The RC low-pass filter for an envelope detector
  in a superheterodyne AM receiver.
• Sample-and-hold circuit
• The capacitor is charged to the voltage of a
  waveform to be sampled.
• The capacitor holds this voltage until an A/D
  converter can convert it to bits.
• The windings in an electric motor or generator
  can be modeled as an RL 1st order circuit.

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First-Order Circuits

• 1st Order Circuit
• One capacitor and one resistor
• The source and resistor may be equivalent to a
  circuit with many resistors and sources.

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First-Order Circuits

• Lets derive the (1st order) differential
  equation for the mesh current i(t).

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First-Order Circuits

• KVL around the loop
• We have

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First-Order Circuits

• The KVL equation becomes
• Differentiating both sides w.r.t. t, we have
• or

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First-Order Circuits

• 1st Order Circuit
• One inductor and one resistor
• The source and resistor may be equivalent to a
  circuit with many resistors and sources.

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First-Order Circuits

• Lets derive the (1st order) differential
  equation for the node voltage v(t).

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First-Order Circuits

• KCL at the top node
• We have

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First-Order Circuits

• The KVL equation becomes
• Differentiating both sides w.r.t. t, we have
• or

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First-Order Circuits

• For all 1st order circuits, the diff. eq. can be
  written as
• The complementary solution is given by
• where K is evaluated from the initial conditions.

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First-Order Circuits

• The time constant of the complementary response
  is t.
• For an RC circuit, t RC
• For an RL circuit, t L/R
• t is the amount of time necessary for an
  exponential to decay to 36.7 of its initial
  value.

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First-Order Circuits

• The particular solution vp(t) is usually a
  weighted sum of f(t) and its first derivative.
• If f(t) is constant, then vp(t) is constant.
• If f(t) is sinusoidal, then vp(t) is sinusoidal.